Main Article Content
On countably uniform closed-spaces
Abstract
Let X be a topological space and Cc(X) be the functionally countable subalgbera of C(X). We call X to be a countably uniform closed-space, brie y, a CUC-space, if Cc(X) is closed under uniform convergence. We investigate that countably uniform closedness need not closed under finite intersection and infinite product. It is shown that if X is a countable union of quasi-components, then X is a CUC-space. We characterize Cc-embedding and also C*c -embedding in CUC- spaces. A subset S of X is called Zc-embedded, if each Z ∈ Zc(S) is the restriction of a zero-set of Zc(X). It is observed that in a zero-dimensional CUC-space, each Lindelof subspae is Zc-embedded. Moreover, it is shown that in CUC-spaces, each Lindelof subspace is Cc-embedded if and only if it is c-completely separated from each zero-set, which is disjoint from it. Also in latter spaces, it is observed that for each S ⊆ X, Cc-embedding, C*c -embedding and Zc-embedding coincide, when S belongs to Zc(X) or it is a c-pseudocompact space. Finally, when X is both a CUC-space and a CP-space, then each Zc-embedded subspace is Cc-embedded (C*c -embedded) in X.
Mathematics Subject Classification (2010): Primary: 54C30, 54C40, 54C05; Secondary: 13C11, 16H20.
Keywords: Functionally countable subalgebra, CUC-space, quasi-component, Zc-embedded, Cc-embedded, C*c -embedded, c-pseudocompact space, CP-space
